Theorems · Theorem · probability

ProbabilityTheory.Kernel.tendsto_densityProcess_limitProcess

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} [inst : MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)}
  {ν : ProbabilityTheory.Kernel α γ},
  κ.fst ≤ ν →
    ∀ [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β},
      MeasurableSet s →
        ∀ᵐ (x : γ) ∂ν a,
          Filter.Tendsto (fun n => κ.densityProcess ν n a x s) Filter.atTop
            (nhds
              (MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
                (ProbabilityTheory.countableFiltration γ) (ν a) x))

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

2 results in Mathlib

Foundations

Depth 319 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel